Analytic description of critical-point actinides in a transition from octupole deformation to octupole vibrations

Dennis Bonatsos, D. Lenis, N. Minkov, D. Petrellis, and P. Yotov

Phys. Rev. C 71, 064309 – Published 22 June 2005

Abstract

An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter ( $ϕ_{0}$ ), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and $B (EL)$ ratios, which depend only on $ϕ_{0}$ , draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well ${}^{226}{Th}$ and ${}^{226}{Ra}$ , for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with $ϕ_{0} = 45^{°}$ for ground-state band spectra and $B (E 2)$ transition rates to the predictions of the X(5) model is pointed out. Analytic solutions are also obtained for Davidson potentials of the form $β^{2} + β_{0}^{4} / β^{2}$ , leading to the AQOA spectrum through a variational procedure.

Received 8 February 2005

DOI:https://doi.org/10.1103/PhysRevC.71.064309

Authors & Affiliations

Dennis Bonatsos^1,*, D. Lenis^1,†, N. Minkov^2,‡, D. Petrellis^1,§, and P. Yotov^2,∥

¹Institute of Nuclear Physics, N.C.S.R. “Demokritos,” GR-15310 Aghia Paraskevi, Attiki, Greece
²Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigrad Road, BG-1784 Sofia, Bulgaria

^*Electronic address: bonat@inp.demokritos.gr
^†Electronic address: lenis@inp.demokritos.gr
^‡Electronic address: nminkov@inrne.bas.bg
^§Electronic address: petrellis@inp.demokritos.gr
^∥Electronic address: pyotov@inrne.bas.bg

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Vol. 71, Iss. 6 — June 2005

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Figure 1
(Color online) (a) Energy ratios $R (L) = E (L) / E (2_{1}^{+})$ as a function of the angle $ϕ_{0}$ . The minima appear as follows: $R (4)$ at $41 . 19^{°}$ , $R (12)$ at $42 . 03^{°}, R (20)$ at $42 . 61^{°}, R (0_{2})$ at $34 . 99^{°}$ , and $R (0_{3})$ at $34 . 13^{°}$ . (b) Same as (a) but for the $B (E 2)$ ratios $R {42} = B (E 2; 4^{+} \to 2^{+}) / B (E 2; 2^{+} \to 0^{+})$ ( $40 . 5^{°}$ ), $R {1210} = B (E 2; 12^{+} \to 10^{+}) / B (E 2; 2^{+} \to 0^{+})$ ( $41 . 0^{°}$ ), $R {2018} = B (E 2; 20^{+} \to 18^{+}) / B (E 2; 2^{+} \to 0^{+})$ ( $42 . 0^{°}$ ), $R {53} = B (E 2;$ $5^{-} \to 3^{-}) / B (E 2; 3^{-} \to 1^{-})$ ( $40 . 5^{°}$ ), $R {1311} = B (E 2; 13^{-} \to$ $11^{-}) / B (E 2; 3^{-} \to 1^{-})$ ( $41 . 5^{°}$ ) and $R {2119} = B (E 2; 21^{-} \to 19^{-}) / B (E 2; 3^{-} \to 1^{-})$ ( $42 . 0^{°}$ ). After each ratio, the position of the maximum (with accuracy $\pm 0 . 5^{°}$ ) appears in parentheses. (c) Same as (a) but for the $B (E 1)$ ratios $R {21} = B (E 1; 2^{+} \to 1^{-}) / B (E 1; 1^{-} \to 0^{+})$ ( $40 . 0^{°}$ ), $R {1211} = B (E 1; 12^{+} \to 11^{-}) / B (E 1; 1^{-} \to 0^{+})$ ( $40 . 5^{°}$ ), $R {2019} = B (E 1; 20^{+} \to 19^{-}) / B (E 1; 1^{-} \to 0^{+})$ ( $41 . 5^{°}$ ), and for the $B (E 3)$ ratios $R {41} = B (E 3; 4^{+} \to 1^{-}) / B (E 3; 3^{-} \to 0^{+})$ ( $41 . 0^{°}$ ), $R {129} = B (E 3; 12^{+} \to 9^{-}) / B (E 3; 3^{-} \to 0^{+})$ ( $41 . 0^{°}$ ), and $R {2017} = B (E 3; 20^{+} \to 17^{-}) / B (E 3; 3^{-} \to 0^{+})$ ( $41 . 5^{°}$ ). Again, each ratio is followed by the position of the maximum (with accuracy $\pm 0 . 5^{°}$ ) in parentheses.Reuse & Permissions
Figure 2
(Color online) (a) Experimental energy ratios $R (L) = E (L) / E (2_{1}^{+})$ for ${}^{220}{Th}$ [27], ${}^{222}{Th}$ [28], ${}^{224}{Th}$ [29], ${}^{226}{Th}$ [30], ${}^{228}{Th}$ [31], ${}^{230}{Th}$ [32], ${}^{232}{Th}$ [32, 33], and ${}^{234}{Th}$ [32], compared to theoretical predictions for $ϕ = 45^{°}$ and $ϕ = 60^{°}$ . (b) Same as (a) but for ${}^{218}{Ra}$ [34, 35], ${}^{220}{Ra}$ [27], ${}^{222}{Ra}$ [32, 36], ${}^{224}{Ra}$ [32, 36], ${}^{226}{Ra}$ [32, 36], and ${}^{228}{Ra}$ [32], compared to theoretical predictions for $ϕ = 45^{°}$ and $ϕ = 56^{°}$ .Reuse & Permissions
Figure 3
(Color online) (a) Experimental energy levels of the ground-state bands of ${}^{220}{Th}$ [27], ${}^{222}{Th}$ [28], ${}^{224}{Th}$ [29], ${}^{226}{Th}$ [30], ${}^{228}{Th}$ [31], ${}^{230}{Th}$ [32], ${}^{232}{Th}$ [32, 33], and ${}^{234}{Th}$ [32], as a function of the neutron number. (b) Same as (a) but for the associated negative-parity bands. (c) Same as (a) but for ${}^{218}{Ra}$ [34, 35], ${}^{220}{Ra}$ [27], ${}^{222}{Ra}$ [32, 36], ${}^{224}{Ra}$ [32, 36], ${}^{226}{Ra}$ [32, 36], ${}^{228}{Ra}$ [32], and ${}^{230}{Ra}$ [32]. (d) Same as (c) but for the associated negative-parity bands.Reuse & Permissions
Figure 4
(Color online) (a) Experimental energy staggering $Δ E$ [Eq. (33)], normalized to the $2_{1}^{+}$ state of each nucleus, for ${}^{220}{Th}$ [27], ${}^{222}{Th}$ [28], ${}^{224}{Th}$ [29], ${}^{226}{Th}$ [30], ${}^{228}{Th}$ [31], ${}^{230}{Th}$ [32], ${}^{232}{Th}$ [32, 33], and ${}^{234}{Th}$ [32], as a function of the angular momentum. (b) Same as (a) but for ${}^{218}{Ra}$ [34, 35], ${}^{220}{Ra}$ [27], ${}^{222}{Ra}$ [32, 36], ${}^{224}{Ra}$ [32, 36], ${}^{226}{Ra}$ [32, 36], ${}^{228}{Ra}$ [32], and ${}^{230}{Ra}$ [32].Reuse & Permissions
Figure 5
(Color online) (a) Experimental $B (E 1; L \to L - 1) / B (E 2; L \to L - 2)$ ratios (multiplied by $10^{5}$ ) [20] of $B (E 1)$ and $B (E 2)$ values originating from the same level of ${}^{226}{Th}$ , compared to theoretical predictions by Bizzeti and Bizzeti-Sona [20] (labeled BBS), as well as to predictions of the present work for $ϕ_{0} = 45^{°}, 60^{°}$ . The ratios corresponding to $L = 10$ and 11 have been used for normalization, as in Ref. 20. (b) Experimental $B (E 1; L \to L + 1) / B (E 1; L \to L - 1)$ ratios [40] of $B (E 1)$ values originating from the same level of ${}^{226}{Ra}$ , compared to three different theoretical predictions from Ref. 39 (labeled R-h, R-I, R-II), as well as to theoretical predictions of the present work for $ϕ_{0} = 45^{°}, 56^{°}$ . See Sec. 4 for further discussion.Reuse & Permissions
Figure 6
(Color online) Energy ratios $R (L) = E (L) / E (2)$ for the ground-state band and the associated negative-parity band ( $s = 1$ ) of the Davidson potentials of Eq. (16), selected through the variational procedure of Sec. 5 (labeled var), compared to the ratios provided by the present work (labeled oct), as a function of angular momentum. The limiting cases corresponding to Davidson potentials with $β_{0} = 0$ (labeled $β_{0} = 0$ ) and $β_{0} \to \infty$ (labeled rot) are also shown for comparison. In all cases, $ϕ_{0} = 45^{°}$ has been used. See Sec. 5 for further discussion.Reuse & Permissions

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